| L(s) = 1 | + (−0.669 + 0.743i)2-s + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (−2.39 + 0.508i)5-s + (0.309 − 0.951i)6-s + (−0.0798 − 2.64i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (1.22 − 2.11i)10-s + (3.08 + 1.22i)11-s + (0.5 + 0.866i)12-s + (−0.137 − 0.422i)13-s + (2.01 + 1.71i)14-s + (1.97 − 1.43i)15-s + (−0.978 + 0.207i)16-s + (2.71 + 3.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (−0.527 + 0.234i)3-s + (−0.0522 − 0.497i)4-s + (−1.06 + 0.227i)5-s + (0.126 − 0.388i)6-s + (−0.0301 − 0.999i)7-s + (0.286 + 0.207i)8-s + (0.223 − 0.247i)9-s + (0.386 − 0.669i)10-s + (0.929 + 0.369i)11-s + (0.144 + 0.249i)12-s + (−0.0380 − 0.117i)13-s + (0.539 + 0.457i)14-s + (0.510 − 0.370i)15-s + (−0.244 + 0.0519i)16-s + (0.658 + 0.731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.638902 + 0.397833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.638902 + 0.397833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 7 | \( 1 + (0.0798 + 2.64i)T \) |
| 11 | \( 1 + (-3.08 - 1.22i)T \) |
| good | 5 | \( 1 + (2.39 - 0.508i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (0.137 + 0.422i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.71 - 3.01i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.120 - 1.14i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-3.35 - 5.80i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.24 + 0.905i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.46 - 0.310i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-7.02 - 3.12i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (4.87 + 3.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + (0.282 - 2.69i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-9.63 - 2.04i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.420 - 3.99i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-14.9 + 3.17i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.131 - 0.227i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.53 + 4.72i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.360 - 3.42i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.501 - 0.556i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (5.43 - 16.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (7.45 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.89 + 8.92i)T + (-78.4 + 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18371238084774721530334631084, −10.25895372920583376878179002343, −9.541755453682123100162201125157, −8.301391213786215586704834132185, −7.44686341322968797390706321925, −6.83513750913576824884148234228, −5.69300216589811041441053112265, −4.35433991966385612183226606280, −3.62943671070504306043091441881, −1.08314118265609672879766042616,
0.78344746593634551737774187691, 2.60863468646841396675525563736, 3.92930397779967804675021579737, 5.05129448746743495359207318913, 6.32890178896087160985508688878, 7.33273538638419442089371040681, 8.367729759932379581076361515076, 8.999051524040015004909497522422, 10.02222852289897394300988260005, 11.26719891120985203768655925466
